{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T03:10:44Z","timestamp":1772766644788,"version":"3.50.1"},"reference-count":29,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T00:00:00Z","timestamp":1685923200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004242","name":"Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia","doi-asserted-by":"publisher","award":["PNURSP2023R368"],"award-info":[{"award-number":["PNURSP2023R368"]}],"id":[{"id":"10.13039\/501100004242","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>It is well known that ranked set sampling (RSS) is more efficient than simple random sampling (SRS). Furthermore, the presence of missing data vitiates the conventional results. Only a minuscule amount of work has been conducted under RSS with missing data. This paper makes a modest attempt to provide some efficient difference- and ratio-type imputation methods in the presence of missing values under RSS. The envisaged imputation methods are demonstrated to provide better results than the existing imputation methods. The theoretical results are enhanced by a computational analysis using real and hypothetically generated symmetric (Normal) and asymmetric (Gamma and Weibull) populations. The computational results show that the proposed imputation method outperforms the existing imputation methods in terms of its higher percent relative efficiency. Additionally, the impact of skewness and kurtosis on the efficiency of the suggested imputation methods has also been calculated.<\/jats:p>","DOI":"10.3390\/axioms12060558","type":"journal-article","created":{"date-parts":[[2023,6,6]],"date-time":"2023-06-06T02:08:15Z","timestamp":1686017295000},"page":"558","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Efficient Difference and Ratio-Type Imputation Methods under Ranked Set Sampling"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3931-888X","authenticated-orcid":false,"given":"Shashi","family":"Bhushan","sequence":"first","affiliation":[{"name":"Department of Statistics, University of Lucknow, Lucknow 226007, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2775-6548","authenticated-orcid":false,"given":"Anoop","family":"Kumar","sequence":"additional","affiliation":[{"name":"Department of Statistics, Amity University, Lucknow 226028, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8780-3655","authenticated-orcid":false,"given":"Tolga","family":"Zaman","sequence":"additional","affiliation":[{"name":"Department of Statistics, Faculty of Science, \u00c7ankiri Karatekin University, \u00c7ankiri 18100, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6619-1725","authenticated-orcid":false,"given":"Aned","family":"Al Mutairi","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1093\/biomet\/63.3.581","article-title":"Inference and missing data","volume":"63","author":"Rubin","year":"1976","journal-title":"Biometrika"},{"key":"ref_2","first-page":"231","article-title":"Experiments with variance estimation from survey data with imputed values","volume":"10","author":"Lee","year":"1994","journal-title":"J. Off. Stat."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1007\/s001840000054","article-title":"Compromised imputation in survey sampling","volume":"51","author":"Singh","year":"2000","journal-title":"Metrika"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1007\/BF02926010","article-title":"Imputation by power transformation","volume":"44","author":"Singh","year":"2003","journal-title":"Stat. Pap."},{"key":"ref_5","first-page":"499","article-title":"A new method of imputation in survey sampling","volume":"43","author":"Singh","year":"2009","journal-title":"Stat. A J. Theor. Appl. Stat."},{"key":"ref_6","first-page":"1727","article-title":"Optimal method of imputation in survey sampling","volume":"3","author":"Singh","year":"2009","journal-title":"Appl. Math. Sci."},{"key":"ref_7","first-page":"207","article-title":"Distinguishing \u2018Missing at Random\u2019 and \u2018Missing Completely at Random\u2019","volume":"50","author":"Heitjan","year":"1996","journal-title":"Am. Stat."},{"key":"ref_8","first-page":"1247","article-title":"Estimation of a population mean using different imputation methods","volume":"7","author":"Ahmed","year":"2006","journal-title":"Stat. Transit."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2226","DOI":"10.1080\/03610920701855020","article-title":"Estimators for the population mean in the case of missing data","volume":"37","author":"Kadilar","year":"2008","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3245","DOI":"10.1080\/03610920903009400","article-title":"Improved estimators of the population mean for missing data","volume":"39","author":"Diana","year":"2010","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_11","first-page":"755","article-title":"Optimal imputation of missing data for estimation of population mean","volume":"19","author":"Bhushan","year":"2016","journal-title":"J. Stat. Manag. Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2576","DOI":"10.1080\/03610926.2016.1167906","article-title":"Optimality of ratio type estimation methods for population mean in presence of missing data","volume":"47","author":"Bhushan","year":"2018","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_13","first-page":"1","article-title":"On optimality of imputation methods for estimation of population mean using higher order moments of an auxiliary variable","volume":"49","author":"Bhushan","year":"2018","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"6588","DOI":"10.1080\/03610918.2016.1208235","article-title":"Imputation using higher order moments of an auxiliary variable","volume":"46","author":"Mohamed","year":"2016","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Bhushan, S., Kumar, A., Pandey, A.P., and Singh, S. (2022). Estimation of population mean in presence of missing data under simple random sampling. Commun. Stat. Simul. Comput., 1\u201322.","DOI":"10.1080\/03610918.2021.2006713"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Anas, M.M., Huang, Z., Shahzad, U., Zaman, T., and Shahzadi, S. (2022). Compromised imputation based mean estimators using robust quantile regression. Commun. Stat. Theory Methods, 1\u201316.","DOI":"10.1080\/03610926.2022.2108057"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1071\/AR9520385","article-title":"A method of unbiased selective sampling using ranked set","volume":"3","author":"McIntyre","year":"1952","journal-title":"Aust. J. Agric. Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02911622","article-title":"On unbiased estimates of the population mean based on the sample stratified by means of ordering","volume":"20","author":"Takahasi","year":"1968","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_19","first-page":"30","article-title":"Ranked set estimation with imputation of the missing observations: The median estimator","volume":"32","year":"2011","journal-title":"Rev. Investig. Oper."},{"key":"ref_20","unstructured":"Bouza, C.N. (2013). Handling Missing Data in Ranked Set Sampling, Springer."},{"key":"ref_21","first-page":"67","article-title":"Ratio estimators of the population mean with missing values using ranked set sampling","volume":"26","author":"Bouza","year":"2014","journal-title":"Environmetrics"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"704","DOI":"10.1080\/15598608.2018.1460886","article-title":"A class of ratio type estimators for imputing the missing values under rank set sampling","volume":"12","author":"Sohail","year":"2018","journal-title":"J. Stat. Theory Pract."},{"key":"ref_23","unstructured":"Arnold, B.C., Balakrishnan, N., and Nagaraja, H.N. (1993). A First Course in Order Statistics, Wiley."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1002\/bimj.200390007","article-title":"Ratio estimators in stratified random sampling","volume":"45","author":"Kadilar","year":"2003","journal-title":"Biom. J."},{"key":"ref_25","unstructured":"Sarndal, C.E., Swensson, B., and Wretman, J. (2003). Model Assisted Survey Sampling, Springer."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Singh, S. (2003). Advanced Sampling Theory with Applications: How Michael Selected Amy, Kluwer. Volumes 1 and 2.","DOI":"10.1007\/978-94-007-0789-4"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1007\/PL00020899","article-title":"An alternative survey in multi-character survey","volume":"48","author":"Singh","year":"1998","journal-title":"Metrika"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"545","DOI":"10.2307\/2556166","article-title":"Ranked set sampling theory with order statistics background","volume":"28","author":"Dell","year":"1972","journal-title":"Biometrics"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1007\/s13571-021-00265-y","article-title":"Novel log type class of estimators under ranked set sampling","volume":"84","author":"Bhushan","year":"2022","journal-title":"Sankhya B"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/6\/558\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:48:31Z","timestamp":1760125711000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/6\/558"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,5]]},"references-count":29,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["axioms12060558"],"URL":"https:\/\/doi.org\/10.3390\/axioms12060558","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,5]]}}}